Precision to calculate Pi

Question

I am trying to calculate pi with Taylor series: pi = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...
But I have to find out how many terms I have to use to limit the error after a specific number of decimals. For example, how many terms in Taylor series should I use to have precision until the 10# decimal place in number pi?
What is the formula for the error in this case?

Answer 1

DAC
I found out the error formula to calculate Pi. The formula is:

error = 4/(2*k+1),

Where k is the number of terms in the series.
I also tested an algorithm in java as follows.

import java.lang.Math;

public class Program
{
public static void main(String[] args) {

double pi=0;
long k = 10000000;
for(long i=0; i<k; i++){
pi += 4*Math.pow(-1,i)/(2*i + 1);
}
double error = 4./(2*k+1);
System.out.println(pi);
System.out.println(error);
System.out.format("%.15f", error);
}
}

https://code.sololearn.com/c2TmRPtDM3iE/?ref=app